Myanmar Matriculation Exam (2016 D)

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2016
MATRICULATION EXAMINATION
DEPARTMENT OF MYANMAR EXAMINATION
MATHEMATICS Time Allowed : (3) Hours
WRITE YOUR ANSWERS IN THE ANSWER BOOKLET.

SECTION (A)

(Answer ALL questions. Choose the correct or the most appropriate answer for each question. Write the letter of the correct or the most appropriate answer.)

1. (1) A function $f$ is defined on the set of real numbers by $f: x \mapsto \frac{3}{x-2}$, $x \neq k$. Then the value of $k$ is

A. 3

B. 1

C. 2

D. $-1$

E. $-3$

(2) An operation $\odot$ is defined by $x \odot y=\frac{3 x y}{x+y}$, then the value of $x$ for which $x \odot 2 x=4$ is

A. $-3$

B. 3

C. 1

D. $-1$

E. 2

(3) $x^{3}-3 x^{2}+k x+7$ is divided by $x+3$, the remainder is 1 . Then $k=$

A. 16

B. 15

C. $-16$

D. $-15$

E. 13

(4) If $x-p$ is a factor of $4 x^{3}-(3 p+2) x^{2}-\left(p^{2}-1\right) x+3$, then $p=$

A. $-\frac{1}{2}$ or 3

B. $\frac{1}{2}$ or $-3$

C. $-1$ or $\frac{3}{2}$

D. 1 or $-\frac{3}{2}$

E. $-1$ or $\frac{2}{3}$

(5) In the expansion of $(3+k x)^{9}$, the coefficients of $x^{3}$ and $x^{4}$ are equal. Then $k=$

A. 1

B. 2

C. 3

D. $-1$

E. $-2$

(6) ${ }^{n} C_{0}+{ }^{n} C_{n-1}=$

A. 0

B. 1

C. 2

D. $n+1$

E. $n$

(7) The solution set in $R$ for the inequation $(x+2)^{2}>2 x+7$ is

A. $\{x \mid x>-3\}$

B. $\{x \mid x<1\}$

C. $\varnothing$

D. $R$

E. $\{x \mid x<-3$ or $x>1\}$

(8) If $p^{\text {th }}$ term of an A.P. is $q$, and the $q^{\text {th }}$ term is $p$, then the common difference is

A. 0

B. 1

C. $-1$

D. 2

E. $-2$

(9) Three positive consecutive terms of a G.P. are $x+1, x+5$ and $2 x+4$ Then $x=$

A. 2

B. 7

C. 3

D. 4

E. 1

(10) If $x, y, 2 x$ is an A.P. and $3,9, y$ is a G.P., then $x+y=$

A. 45

B. 54

C. 27

D. 9

E. $-9$

(11) $A=\left(\begin{array}{cc}2 & 0 \\ 1 & 5\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ 2 & k\end{array}\right)$. Then the value of $k$ for which $A B=B A$ is

A. $-1$

B. 1

C. 7

D. $-4$

E. 4

(12) Given that $A$ is a $2 \times 2$ matrix such that

$\left(\begin{array}{cc}2 & -1 \\ 3 & 4\end{array}\right) A+\left(\begin{array}{cc}1 & 1 \\ -3 & -1\end{array}\right) A=\left(\begin{array}{cc}3 & 6 \\ -3 & 9\end{array}\right)$, then the matrix $A$ is

$A \cdot\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)$

B. $\left(\begin{array}{cc}1 & 2 \\ -1 & 3\end{array}\right)$

C. B. $\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)$

D. $\left(\begin{array}{cc}2 & -1 \\ -1 & 0\end{array}\right)$

E. $\left(\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right)$

(13) If $A$ is an event such that $P(A)=x$ and $P(\operatorname{not} A)=y$, then $x^{3}+y^{3}=$

A. $3 x y$ 

B. $1+3 x y$

C. $3 x y-1$

D. $1-3 x y \quad$ 

E. none of these

(14) In 100 trials, $A$ is an event and the expected frequency of $A$ is 30 , then $P(A)=$

A. $\frac{3}{10}$

B. $\frac{3}{5}$

C. $\frac{3}{20}$

D. $\frac{1}{30}$

E. $\frac{1}{100}$

(15) In $\odot O, D C / / A B$ and $\angle C A B=20^{\circ}$, Then $\angle D A C=$ 


A, $20^{\circ}$

B. $15^{\circ}$

C,.$50^{\circ}$

D. $30^{\circ}$

E. $40^{\circ}$

(16) $A$ and $B$ are two points on a circle $3 \mathrm{~cm}$ apart. The chord $A B$ is produced to $C$ making $B C=1 \mathrm{~cm}$, Then the length of the tangent from $C$ to the circle is

A. $2 \mathrm{~cm}$

B. $1 \mathrm{~cm}$

C, $3 \mathrm{~cm}$

D.  4 cm

E.  5 cm

(17) In the trapezium $A B C D, A B$ is twice $D C$ and $A B / / D C$. If $A C$ and $B D$ intersect at $O$, then $\alpha(\triangle A O B): \alpha(\triangle C O D)=$

A. $1: 4$

B. $2: 3$

C. $4: 1$

D. $3: 2$

E. none of these

(18) If $\vec{a}, \vec{b}$ are non-parallel and non-zero such that $(3 x+y) \vec{a}+(y-3) \vec{b}=\overrightarrow{0}$, then $x=$

A. 1

B, $-1$

C. 3

D. $-3$

E. none of these

(19) If $P=(3,4), R=(8,2)$ and $O$ is the orgin and $\overrightarrow{O P}=\overrightarrow{O T}-\frac{1}{2} \overrightarrow{O R}$, then the coordinates of the point $T$ is

A. $(1,3)$

B. $(2,4)$

C, $(7,5)$

D. $(4,5)$

E. $(5,7)$

(20) What is the smallest value of $x$ for which $\tan 3 x=-1$ ?

A. $15^{\circ}$

B. $45^{\circ}$

C. $75^{\circ}$

D. $90^{\circ}$

E, $105^{\circ}$

(21) If $A, B, C$ are she angles of a triangle and $\tan A=3$ and $\tan B=2$, then $\tan C=$

A. 1

B. 2

C. 3

D. 4

E. 5

(22) If $\sin 20^{\circ}=p$, then $\sec 70^{\circ}=$

A. $p$

B. $2 p$

C. $-p$

D. $\frac{1}{p}$

E. none of these

(23) If $f(x)=1-\frac{1}{x}$, then $f^{\prime}\left(\frac{1}{2}\right)=$

A. 2

B. 3

C. 4

D. 5

E. 6

(24) If $V=\frac{4}{3} r^{3}-\frac{3}{4} r^{2}+r-5$, then the rate of change of $V$ with respect to $r$ when $r=2$ is

A. 6

B. 7

C. 8

D. 9

E. 14

(25) The gradient of normal line to the curve $y=2 \sqrt{x}$ at the point $x=9$ is

A. $\frac{1}{3}$

B. $-\frac{1}{3}$

C. 3

D. $-3$

E. 6



SECTION B

(Answer ALL questions)

2. The function $f$ is defined, for $x \in R$, by $f(x)=2 x-3$. Find the value of $x$ for which $f(x)=f^{-1}(x)$

(OR)

Find the value of $k$ if $4 x^{7}+5 x^{3}-2 k x^{2}+7 k-4$ has a remainder of 12 when divided by $x+1$

3. The ninth term of an arithmetic progression is 6 . Find the sum of the first 17 terms.

(OR)

A geometric progression is such that the sum of the first 3 terms is $0.973$ times the sum to infinity. Find the common ratio.

4. Given : $\odot O$ with $A B=A D$ and $A C$ is a diameter. 


$$\text { Prove : } B C=C D$$

5. Given that $A=B+C$, prove that $\tan A-\tan B-\tan C=\tan A \tan B \tan C$.

6. Differentiate $y=\frac{1}{x}$ with repsect to $x$ from the first principles.


SECTIONC

(Answer any SIX questions)

7. (a) Functions $f$ and $g$ are defined by $f(x)=\frac{x}{2-x}, x \neq 2$ and $g(x)=a x+b$. Given that $g^{-1}(7)=3$ and $(g \circ f)(5)=-7$, calculate the value of $a$ and of $b$.

(b) A binary operation $\odot$ on $R$ is defined by $x \odot y=x^{2}-2 x y+2 y^{2}$. Find $(3 \odot 2) \odot 4$. If $(3 \odot k)-(k \odot 1)=k+1$, find the values of $k$


8. (a) The cubic polynomial $f(x)$ is such that the coefficient of $x^{3}$ is $-1$ and the roots of the equation $f(x)=0$ are 1,2 and $k$. Given that $f(x)$ has a remainder of 8 when divided by $x-3$, find the value of $k$ and the remainder when $f(x)$ is divided by $x+3$

(b) The expansion of $(3+4 x)^{n}$, the coefficients of $x^{4}$ and $x^{5}$ are in the ratio of $5: 16$. Find the value of $n$


9. (a) Find the solution set in $R$ of the inequation $(x-6)^{2}>x$ by graphical method and illustrate it on the number line.

(b) The third term of an A.P. is 9 and the seventh term is $49 .$ Calulate the thirteenth term. Which term of the progression, if any, is $289 ?$


10. (a) The first and second terms of a G.P. are 10 and 11 respectively. Find the least number of terms such that their sum exceeds 8000 .

(b) The matrices $A$ and $B$ are such that $A=\left(B^{-1}\right)^{2}$. Given that $B=\left(\begin{array}{cc}2 & -1 \\ 2 & 1\end{array}\right)$ find the value of the constant $k$ for which $k B^{-1}=4 A+I$, where $I$ is the identity matrix of order 2 .


11. (a) Given that $A=\left(\begin{array}{cc}4 & -1 \\ -3 & 2\end{array}\right)$, use the inverse matrix of $A$ to solve the simultaneous equations $y-4 x+8=0,2 y-3 x+1=0$.

(b) Three tennis players $A, B, C$ play each other only once. The probability that $A$ will beat $B$ is $\frac{2}{7}$, that $B$ will beat $C$ is $\frac{1}{3}$ and that $C$ will beat $A$ is $\frac{2}{5}$. Calculate the probability that $A$ wins both games.


12. (a) Prove that the opposite angles of a quadrilateral inscribed in a circle are supplementary.

(b) $A B C D$ is a parallelogram. Any circle through $A$ and $B$ cuts $D A$ and $C B$ produced at $P$ and $Q$ respectively. Prove that $D C Q P$ is cyclic.


13. (a) In the figure, $A B / / C D$ and $\alpha(\triangle E C D): \alpha(A B D C)=16: 9$. Find the numerical value of $C D: A B$. 


Given that $\alpha(\triangle E C D)=24 \mathrm{~cm}^{2}$, calculate $\alpha(\Delta E A B)$.

(b) The position vectors of the points $A,B$ and $C$, relative to an origin $O$, are 

 $2 \hat{i}+3 \hat{j}, 10 \hat{i}+2 \hat{j}$ and $\lambda(-\hat{i}+5 \hat{j})$ respectively. Given that $|\overrightarrow{A B}|=|\overrightarrow{A C}|$ show that $\lambda^{2}-\lambda-2=0$ and hence find the two possible vectors $\overrightarrow{A C}$


14. (a) If $\cot x+\cos x=p$ and $\cot x-\cos x=q$, show that $\sqrt{p q}=\cos x \cot x$, where $x$ is acute and hence, prove that $p^{2}-q^{2}=4 \sqrt{p q}$

(b) A man travels $10 \mathrm{~km}$ in a direction $\mathrm{N} 70^{\circ} \mathrm{E}$ and then $5 \mathrm{~km}$ in a direction N $40^{\circ} \mathrm{E}$. What is his final distance and bearing from his starting point?


15. (a) If $y=(3+4 x) e^{-2 x}$, then prove that $\frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+4 y=0$.

(b) Find the minimum value of the sum of a positive number and its reciprocal.


Answer

1)

1 C

2 E

3 C

4 C

5 B

6 D

7 E

8 C

9 B

10 A

11 C

12 B

13 D

14 A

15 C

16 A

17 C

18 B

19 C

20

21 A

22 D

23 C

24 E

25 D


2) $x=3[O R] \quad k=5$

3) $S_{17}=102[O R] \quad r=0.3$

4) Prove

5) Prove

6) $\frac{d y}{d x}=-\frac{1}{x^{2}}$

7) (a) $a=3, b=-2 \quad$ (b) $k=2$ or 3

8) (a) $k=7, R=200$

(b) $n=16$

9) $(a)\{x \mid x<4$ or $x>9\}$ (b) $u_{13}=109,n=31$

10) $(a) 47$ (b) $k=3$

11) (a) $x=3, y=4$ (b) $\frac{6}{35}$

12) (a)  Prove (b) prove  

13)  (a) $\alpha(\Delta E A B)=37.5 \mathrm{~cm}^{2}$

(b) $\overrightarrow{A C}=-4 \hat{i}+7 \hat{j} \text { or } \overrightarrow{A C}=-\hat{i}-8 \hat{j}$

14) (a) Prove (b) $14.55 \mathrm{~km}, \mathrm{~N} 60^{\circ}{6}^{\prime} \mathrm{E}$

15) (a) Prove (b) 2


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